proof of ratio test

Assume $k<1$ . By definition $\exists N$ such that
$n>N\implies|\frac{a_{n+1}}{a_{n}}-k|<\frac{1-k}{2}\implies|\frac{a_{n+1}}{a_{n% }}|<\frac{1+k}{2}<1$

i.e. eventually the series $|a_{n}|$ becomes less than a convergent geometric series, therefore a shifted subsequence of $|a_{n}|$ converges by the comparison test. Note that a general sequence $b_{n}$ converges iff a shifted subsequence of $b_{n}$ converges. Therefore, by the absolute convergence theorem, the series $a_{n}$ converges.

Similarly for $k>1$ a shifted subsequence of $|a_{n}|$ becomes greater than a geometric series tending to $\infty$ , and so also tends to $\infty$ . Therefore $a_{n}$ diverges.

Title	proof of ratio test
Canonical name	ProofOfRatioTest
Date of creation	2013-03-22 12:24:46
Last modified on	2013-03-22 12:24:46
Owner	vitriol (148)
Last modified by	vitriol (148)
Numerical id	6
Author	vitriol (148)
Entry type	Proof
Classification	msc 40A05
Classification	msc 26A06